General differential equation solution for Kepler Problem

[Steve Jones]

To be honest, I don't know any physics. I am a high school student who has taken high school physics, but America's education system isn't known for teaching much more than Newton's laws. I have, however, taken Multivariable/Vector calculus, so I have a decent math background.

I was wondering is there is a specific form of the solution to the Kepler problem. The initial conditions would be the masses, positions, and velocities. I have found this link to the wikipedia solution, but I wonder if it is possible to have a solution that I can just plug the masses, velocities, and positions in and get an equation for the motion of both bodies.

Also, I wonder if that solution could be generalized to include however many bodies you want. The wikipedia article said it could not be solve in terms of first integrals, but I wonder if there is a general solution for n-bodies.

Please be nice to me :P I don't possesses a vast knowledge of physics (or any at all). I also don't know if this thread is in the right place either.

 

[mfb]

I have found this link to the wikipedia solution, but I wonder if it is possible to have a solution that I can just plug the masses, velocities, and positions in and get an equation for the motion of both bodies.

It is possible to make that, but the formula would be too long to be practical. It is easier to calculate all the relevant quantities step by step, as shown in the article and books.

Also, I wonder if that solution could be generalized to include however many bodies you want. The wikipedia article said it could not be solve in terms of first integrals, but I wonder if there is a general solution for n-bodies.

There is not.

 

[Steve Jones]

Thank you for the response.

 

[rumborak]

The problem is that from three bodies on, in general, the trajectories get very erratic. Minute differences between starting points make the difference between a stable situation and one where one of the bodies gets ejected from the system.

 

[AgentSmith]

To be honest, I don't know any physics. I am a high school student who has taken high school physics, but America's education system isn't known for teaching much more than Newton's laws. I have, however, taken Multivariable/Vector calculus, so I have a decent math background.
snipAlso, I wonder if that solution could be generalized to include however many bodies you want. The wikipedia article said it could not be solve in terms of first integrals, but I wonder if there is a general solution for n-bodies.

Please be nice to me :P I don't possesses a vast knowledge of physics (or any at all). I also don't know if this thread is in the right place either.

Hi Steve: You know a heck of a lot more than most high schoolers. Most of them wouldn't know a differential equation from differential fluid.
In classical mechanics courses following general college physics they discuss the "three-body problem". There is no general solution to 3 or more bodies. You can, however, use numerical methods to approximate a solution to whatever degree you want.
I disagree with your statement that you don't know any physics.